Question 1199265
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Solve for x       2x^(2/3) - 5x^(1/3) - 3 = 0
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<pre>
Prepare your equation to factoring using grouping-regrouping &nbsp;(in the same way as you did in your post)

    2x^(2/3) - 6x^(1/3) + x^(1/3) - 3 = 0


Now factor left side step by step

    2x^(1/3)*(x^(1/3)-3) + (x^(1/3) - 3) = 


    (x^(1/3)-3)*(2x^(1/3)+1) = 0.


From this factored form you have EITHER first parentheses expression is zero, 
OR the second parentheses expression is zero.


(a)  Case  x^(1/3)-3 = 0.

     It gives  x^(1/3) = 3,  which implies  x = 3^3 = 27.



(b)  Case  2x^(1/3)+1 = 0.

     It gives  2x^(1/3) + 1  0,  which implies  x^(1/3) = {{{-1/2}}},  x = {{{(-1/2)^3}}} = {{{-1/8}}}.
</pre>

Solved.


The answers/(the solutions) &nbsp;&nbsp;are  &nbsp;&nbsp;x= 27  &nbsp;&nbsp;and/or  &nbsp;&nbsp;x= {{{-1/8}}},  &nbsp;&nbsp;same as in your textbook.


Victory (!)



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In your post, your very first step was correct, but your factoring after that was wrong.